Dedekind cut

Dedekind used his cut to construct the irrational, real numbers.

In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two non-empty parts A and B, such that all elements of A are less than all elements of B and A contains no greatest element. The cut itself is, conceptually, the "gap" defined between A and B. In other words, A contains every rational number less than the cut, and B contains every rational number greater than the cut. The cut itself is in neither set.

More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty parts, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards, and A contains no greatest element. See also completeness (order theory).

The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves. Wherever a cut occurs and it is not on a real rational number, an irrational number (which is also a real number) is created by the mathematician. Through the use of this device, there is considered to be a real number, either rational or irrational, at every point on the number line continuum, with no discontinuity.

Whenever, then, we have to do with a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut ... . From now on, therefore, to every definite cut there corresponds a definite rational or irrational number ....

—Richard Dedekind, Continuity and Irrational Numbers, Section IV

Dedekind used the ambiguous word cut (Schnitt) in the geometric sense. That is, it is an intersection of a line with another line that crosses it. It is not a gap. When one line crosses another in geometry, it is said to cut that line. In this case, one of the lines is the number line. Both lines have one point in common. At that one point on the number line, if there is no rational number, the mathematician posits or arbitrarily places an irrational number. This results in the positioning of a real number at every point on the continuum.

Handling Dedekind cuts

It is more symmetrical to use the (A,B) notation for Dedekind cuts, but each of A and B does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one 'half' — say, the lower one — and call any downward closed set A without greatest element a "Dedekind cut".

If the ordered set S is complete, then every set B in a Dedekind cut (A, B) must have a minimal element b, hence we must have that A is the interval ( −∞, b), and B the interval [b, +∞). In this case, we say that b is represented by the cut (A,B).

The important purpose of the Dedekind cut is to work with numbers that are not complete. The cut itself can represent a number not in the original collection of numbers (most often rational numbers). The cut can represent a number b, even though the numbers contained in the two sets A and B do not actually include the number b that their cut represents.

For example if A and B only contain rational numbers, they can still be cut at √2 by putting every negative rational number in A, along with every non-negative number whose square is less than 2; similarly B would contain every positive rational number whose square is greater than 2. Even though there is no rational value for √2, if the rational numbers are partitioned into A and B this way, the partition itself represents an irrational number.

Ordering Dedekind cuts

Regard one Dedekind cut (A, B) as less than another Dedekind cut (C, D) if A is a proper subset of C. Equivalently, if D is a proper subset of B, the cut (A, B) is again less than (C, D). In this way, set inclusion can be used to represent the ordering of numbers, and all other relations (greater than, less than or equal to, equal to, and so on) can be similarly created from set relations.

The set of all Dedekind cuts is itself a linearly ordered set (of sets). Moreover, the set of Dedekind cuts has the least-upper-bound property, i.e., every nonempty subset of it that has any upper bound has a least upper bound. Embedding S within a larger linearly ordered set that does have the least-upper-bound property is the purpose.

The cut construction of the real numbers

A typical Dedekind cut of the rational numbers is given by

This cut represents the irrational number √2 in Dedekind's construction. Note that the equality b² = 2 cannot hold since √2 is not rational.

Additional structure on the cuts

See: Construction of the real numbers

Generalization: Dedekind completions in posets

More generally, if S is a partially ordered set, a completion of S means a complete lattice L with an order-embedding of S into L. The notion of complete lattice generalizes the least-upper-bound property of the reals.

One completion of S is the set of its downwardly closed subsets (also called order ideals), ordered by inclusion. S is embedded in this lattice by sending each element x to the ideal it generates.

Dedekind-MacNeille completion

A related completion that preserves all existing sups and infs of S is obtained by the following construction: For each subset A of S, let A^u denote the set of upper bounds of A, and let A^l denote the set of lower bounds of A. (These operators form a Galois connection.) Then the Dedekind-MacNeille completion of S consists of all subsets A for which

(A^u)^l = A;

it is ordered by inclusion. The Dedekind-MacNeille completion is generally a smaller lattice than the lattice of order ideals; S is embedded in it in the same way. It is the smallest complete lattice with S embedded in it.

The Dedekind-MacNeille completion of a Boolean algebra is a complete Boolean algebra.

Another generalization: surreal numbers

A construction similar to Dedekind cuts is used for the construction of surreal numbers.

Allusions

In his chapter on Henri Bergson, the author C.E.M. Joad employed imagery that was similar to Dedekind's concept of the cut. Joad was trying to explain how Bergson saw the mind as an instrument that projected permanent objects onto the experience of constant change. "The intellect, then, is a purely practical faculty, which has evolved for the purposes of action. What it does is to take the ceaseless, living flow of which the universe is composed and to make cuts across it, inserting artificial stops or gaps in what is really a continuous and indivisible process. The effect of these stops or gaps is to produce the impression of a world of apparently solid objects. These have no existence as separate objects in reality; they are, as it were, the design or pattern which our intellects have impressed on reality to serve our purposes." This is reminiscent of Dedekind's creation of a new irrational number at every gap in the continuous number line at which there is no existing real number.^[1]

See also

• Cauchy sequence

Notes

1. ^ Great Philosophies of the World, C.E.M. Joad, Ch. VI, "The Philosophy of Change," 1930:Jonathan Cape and Harrison Smith, Inc.

References

• Dedekind, Richard, Essays on the Theory of Numbers, "Continuity and Irrational Numbers," Dover: New York, ISBN 0-486-21010-3. Also available at Project Gutenberg.